Alternative anecdote. I was a students physics TA which involved lecturing and labs and lots of homeworks and in class work. I had a student who really never seemed to understand algebra and he came and said he couldn't really understand how to do the physics problems very well. I said well its going to be hard without really getting your algebra issues figured out and so he just went and did that. I think having a solid understanding of where it is deeply useful (and math and physics have evolved together forever) can be the spark that lights the fire to build the missing understanding. I can say math was very rote for me until I got into physics and I would never have gotten into hairy math without a physics motivation.